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Complex Antiderivative

  1. Feb 2, 2016 #1
    1. The problem statement, all variables and given/known data
    How would one go about finding the antiderivative to this function?
    5c664275712dfef070bc027353aaecd0.png

    2. Relevant equations
    N/A

    3. The attempt at a solution
    This problem has been rather tricky I have tried several attempts at the solution. My one solution consists of me factoring out the x^4. Looking for some guidance please.

    Thank you!
     
  2. jcsd
  3. Feb 2, 2016 #2

    Mark44

    Staff: Mentor

    I would start with an ordinary substitution, u = x2, and would complete the square in the radical. From there, a trig substitution seems promising.
     
  4. Feb 2, 2016 #3
    OK I will work it out now thank you for the suggestion.
     
  5. Feb 2, 2016 #4
    OK I gave your suggestion an attempt.
    I've arrived at the following after the substitution from x ---> u completing the square----> and back to x.

    (x^4-1)
    x^2((x^2+1/4)^2+ (3/4)))^(1/2)
     
  6. Feb 2, 2016 #5

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    Where is the point in substituting back before integration?
    u=x^2+1/2 (not 1/4) was my first idea as well, but then you still have an ugly sqrt(u) in the denominator.
     
  7. Feb 2, 2016 #6
    if you let u= x^2+(1/2) I also have a hard time figuring out how to remove the dx and convert it to du if du= 2x
     
  8. Feb 2, 2016 #7

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    That's where the sqrt(u-1/2) comes in. Forgot the 1/2 in the previous post.
    Hmm, u=x^2 is an easier substitution. The initial square root goes away anyway.

    Thinking about it... standard partial fraction decomposition should work, with imaginary numbers to have the zero.
     
  9. Feb 2, 2016 #8
    Should i apply partial fraction before or after u substitution. Also sorry for still not getting it but when I u substitute i'm still having a hard time figuring out how to change the integral from being with respect to x to respect to u.
     
  10. Feb 3, 2016 #9

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    Well, if u=x2, then x=+-sqrt(u) and du = 2x dx.

    Partial fraction decomposition would be without substitution.
     
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